Achieving Linear Scaling for Parallel Direct Condensation using Disk as Primary Storage

Condensation is a critical technique for the Modular Atlas project, which analyzes modular matrix representations for finite groups. For large simple groups, these representations are more efficient computationally than the next alternative: permutation representations. A new approach to direct condensation is presented that uses parallel disk as primary storage. This approach offers a running time of O(sn), for n the size of the orbit being condensed and s the size of the generating set. This is an improvement over the previous landmarks approach, which offers a running time of O(Lsn). For fixed storage, L, the landmark ratio, increases linearly with n. Hence the previous approach is quadratic in n. With this disk-based approach, we produce condensation matrices for the sporadic simple Janko Group J4 on an orbit of size 131, 358, 148, 251. Each point of the orbit is represented as 4-dimensional subspace in a vector space of dimension 112 over GF (2). This demonstrates a condensation of a group whose orbit is 10 times larger than the largest previous such group for which direct condensation was applied. NOTE TO REVIEWER: We have currently tested our algorithm on the Fischer Group, Fi23. For J4, we have completed the orbit enumeration and the suborbit partitioning. We are currently running the final phase, suborbit counting, and will have results for this in 2 weeks.